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Thermo-rotational buckling and post-buckling analyses of rotating functionally graded microbeams

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Abstract

Buckling and post-buckling responses of rotating clamped–clamped functionally graded microbeams in thermal environment are examined on the basis of the Euler–Bernoulli beam assumption. To enrich the formulation with the size effect the modified couple stress theory is employed. The temperature dependency of material properties is considered. The nonlinear finite element technique alongside with the Newton–Raphson technique is utilized to extract the prestressed deformation. Moreover, the direct iteration method is employed to determine the buckling point and post-buckling equilibrium path. The impacts of material length scale parameter, volume fraction exponent, rotor radius, microbeam length to its thickness proportion and rotation speed on the presented outcomes are examined.

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Abbreviations

\(A_{11}\) :

Axial rigidity

\(B_{11}\) :

Axial-bending rigidity

\(b\) :

Microbeam width

\(D_{11}\) :

Bending rigidity

\(E(z)\) :

FGM Young’s modulus

\(E_{mR}\) :

Metal phase Young’s modulus in reference temperate

\({\mathbf{F}}\) :

Force vector

\({\mathbf{F}}^{e}\) :

kth element force vector

\(h\) :

Microbeam thickness

\(h_{b}\) :

Bottom layer thickness

\(h_{c}\) :

Core layer thickness

\(h_{t}\) :

Top layer thickness

\(I_{0}\) :

Microbeam mass per its length

\(I_{1}\) :

First-order mass moment of inertia

\(I_{2}\) :

Second-order mass moment of inertia

\({\mathbf{K}}_{(1)}\) :

Linear stiffness matrix

\({\mathbf{K}}_{(2)}\) :

Second order stiffness matrix

\({\mathbf{K}}_{(3)}\) :

Third order stiffness matrix

\({\mathbf{K}}^{e}\) :

kth element stiffness matrix

\({\mathbf{K}}_{G}\) :

Geometric stiffness matrix

\({\mathbf{K}}_{NL}\) :

Nonlinear stiffness matrix

\({\mathbf{K}}_{(1)}^{El}\) :

Linear elastic stiffness matrix

\({\mathbf{K}}_{NL}^{EL}\) :

Nonlinear elastic stiffness matrix

\(L\) :

Micro-beam length

\(l\) :

Material length scale parameter

\(l_{e}\) :

Element length

\(M_{xx}\) :

Moment resultant

\(M_{xx}^{T}\) :

Thermal moment resultant

\(m_{xy}\) :

Couple stress moment

\({\mathbf{N}}_{u}\) :

Axial displacement shape function

\({\mathbf{N}}_{w}\) :

Deflection shape function

\(N_{xx}\) :

Axial force resultant

\(N_{xx}^{T}\) :

Thermal force resultant

\(N_{xx}^{T,cr}\) :

Critical thermal force resultant

\(n\) :

Volume fraction exponent

\(P_{c}\) :

Ceramic material property

\(P_{m}\) :

Metal material property

\(P_{xy}\) :

Couple stress moment resultant

\(Q_{11} (z)\) :

FGM elastic stiffness

\({\mathbf{q}}\) :

Microbeam generalized coordinate vector

\({\mathbf{q}}^{e}\) :

Element generalized coordinate vector

\(R\) :

Rotor radius

\({\mathbf{R}}_{NL}\) :

Prestressed residual vector

\(T\) :

Microbeam kinetic energy

\(T_{ref}\) :

Temperature reference (300 K)

\(U\) :

Microbeam strain energy

\(U_{0}\) :

Axial displacement of a typical particle

\(u(x,t)\) :

Axial displacement of a typical particle lied on geometrical middle plane of core layer

\(V_{c}\) :

Volume faction of ceramic phase

\({\mathbf{v}}\) :

Critical buckling eigenvector

\(W\) :

Deflection of a typical particle

\(w(x,t)\) :

Deflection of a typical particle lied on geometrical middle plane of core layer

\(w_{mid}\) :

Deflection of the microbeam mid-point

\(\hat{w}_{mid}\) :

Nondimensional deflection of the microbeam mid-point

\(\alpha (z)\) :

FGM thermal expansion coefficient

\(\Delta T\) :

Temperature increment

\(\Delta T_{Cr}\) :

Critical buckling temperature increment

\(\delta {\mathbf{q}}_{ps}\) :

Incremental prestressed generalized coordinate vector

\(\varepsilon^{T}\) :

Thermal strain

\(\varepsilon_{xx}\) :

Total strain

\(\varepsilon_{xx}^{0}\) :

Membrane strain

\(\varepsilon_{xx}^{1}\) :

Flexural strain (curvature)

\({\rm M}\) :

FGM size-dependent rigidity

\(\mu (z)\) :

FGM shear modulus

\(\nu\) :

FGM Poisson’s ratio

\(\xi\) :

Normalized local axial coordinate for an element

\(\rho (z)\) :

FGM mass density

\(\sigma_{xx}\) :

Cauchy stress

χ xy :

Curvature

\(\hat{\varOmega }\) :

Nondimensional rotation speed

\(\omega_{R}\) :

Rotation speed

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Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for profit sectors.

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Authors and Affiliations

  1. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    Seyyed Mohammad Hossein Hosseini & Hadi Arvin

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  1. Seyyed Mohammad Hossein Hosseini
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  2. Hadi Arvin
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Correspondence to Hadi Arvin.

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Appendix

Appendix

  1. a

    The element stiffness matrix

$${\mathbf{K}}^{e} = \sum\limits_{i = 1}^{15} {{\mathbf{K}}_{i} }$$
(A.1)
$$\begin{aligned} & {\mathbf{K}}_{1} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)A_{11} {\mathbf{N}}_{u}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{u}^{{\prime }} } d\xi ,\quad {\mathbf{K}}_{2} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{3} }}} \right)B_{11} } {\mathbf{N}}_{u}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime \prime }} d\xi , \\ & {\mathbf{K}}_{3} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)N_{xx}^{T} {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} } {\mathbf{N}}_{w}^{{\prime }} d\xi ,\quad {\mathbf{K}}_{4} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{3} }}} \right)B_{11} } {\mathbf{N}}_{w}^{{{\prime \prime }{\text{T}}}} {\mathbf{N}}_{u}^{{\prime }} d\xi , \\ & {\mathbf{K}}_{5} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{4} }}} \right)D_{11} } {\mathbf{N}}_{w}^{{{\prime \prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime \prime }} d\xi ,\quad {\mathbf{K}}_{6} = - (l_{e} )\int_{0}^{1} {I_{0} \omega_{R}^{2} {\mathbf{N}}_{u}^{\text{T}} {\mathbf{N}}_{u} d\xi } , \\ & {\mathbf{K}}_{7} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e} }}} \right)I_{1} \omega_{R}^{2} {\mathbf{N}}_{u}^{\text{T}} {\mathbf{N}}_{w}^{{\prime }} d\xi } ,\quad {\mathbf{K}}_{8} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e} }}} \right)I_{1} \omega_{R}^{2} {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{u} d\xi } , \\ & {\mathbf{K}}_{9} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)I_{2} \omega_{R}^{2} {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime }} d\xi } ,\quad {\mathbf{K}}_{10} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{4} }}} \right){\rm M}{\mathbf{N}}_{w}^{{{\prime \prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime \prime }} d\xi } , \\ & {\mathbf{K}}_{11} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)A_{11} } \frac{du}{dx}{\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime }} d\xi ,\quad {\mathbf{K}}_{12} = \frac{1}{2}(l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)A_{11} } \frac{dw}{dx}{\mathbf{N}}_{u}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime }} d\xi , \\ & {\mathbf{K}}_{13} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{3} }}} \right)B_{11} } \frac{dw}{dx}{\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime \prime }} d\xi ,\quad {\mathbf{K}}_{14} = - \frac{1}{2}(l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{3} }}} \right)B_{11} } \frac{dw}{dx}{\mathbf{N}}_{w}^{{{\prime \prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime }} d\xi , \\ & {\mathbf{K}}_{15} = \frac{1}{2}(l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)A_{11} } (\frac{dw}{dx})^{2} {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} {\mathbf{N}}_{w}^{{\prime }} d\xi , \\ \end{aligned}$$
(A.2)
  1. b

    The kth element force vector

$${\mathbf{F}}^{e} = \sum\limits_{i = 1}^{6} {{\mathbf{F}}_{i} }$$
(A.3)
$$\begin{aligned} & {\mathbf{F}}_{1} = (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e} }}} \right)N_{xx}^{T} } {\mathbf{N}}_{u}^{{{\prime }{\text{T}}}} d\xi ,\quad {\mathbf{F}}_{2} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e}^{2} }}} \right)M_{xx}^{T} } {\mathbf{N}}_{w}^{{{\prime \prime }{\text{T}}}} d\xi ,\quad {\mathbf{F}}_{3} = (l_{e} )\int_{0}^{1} {I_{0} \omega_{R}^{2} R} {\mathbf{N}}_{u}^{\text{T}} d\xi , \\ & {\mathbf{F}}_{4} = (l_{e} )\int_{0}^{1} {I_{0} \omega_{R}^{2} \left( {(k - 1) + \xi } \right)} l_{e} {\mathbf{N}}_{u}^{\text{T}} d\xi ,\quad {\mathbf{F}}_{5} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e} }}} \right)I_{1} \omega_{R}^{2} R} {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} d\xi , \\ & {\mathbf{F}}_{6} = - (l_{e} )\int_{0}^{1} {\left( {\frac{1}{{l_{e} }}} \right)I_{1} \omega_{R}^{2} \left( {(k - 1) + \xi } \right)l_{e} } {\mathbf{N}}_{w}^{{{\prime }{\text{T}}}} d\xi \\ \end{aligned}$$
(A.4)

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Hosseini, S.M.H., Arvin, H. Thermo-rotational buckling and post-buckling analyses of rotating functionally graded microbeams. Int J Mech Mater Des 17, 55–72 (2021). https://doi.org/10.1007/s10999-020-09509-7

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